Make to stock vs. Make to Order

1
Make to stock vs. Make to Order

• Made-to-stock (MTS) operations
• Product is manufactured and stocked in advance of
  demand
• Inventory permits economies of scale and protects
  against stockouts due to variability of inflows
  and outflows
• Make-to-order (MTO) process
• Each order is specific, cannot be stored in
  advance
• Process manger needs to maintain sufficient
  capacity
• Variability in both arrival and processing time
• Role of capacity rather than inventory
• Safety inventory vs. Safety Capacity
• Example Service operations

2
Examples

• Banks (tellers, ATMs, drive-ins)
• Fast food restaurants (counters, drive-ins)
• Retail (checkout counters)
• Airline (reservation, check-in, takeoff, landing,
  baggage claim)
• Hospitals (ER, OR, HMO)
• Service facilities (repair, job shop,
  ships/trucks load/unload)
• Some production systems- to some extend (Dell
  computer)
• Call centers (telemarketing, help desks, 911
  emergency)

3
The DesiTalk Call Center
The Call Center Process
Sales Reps Processing Calls (Service Process)
Incoming Calls (Customer Arrivals)
Answered Calls (Customer Departures)
Calls on Hold (Service Inventory)
Blocked Calls (Due to busy signal)
Abandoned Calls (Due to long waits)
Calls In Process (Due to long waits)
4
Service Process Attributes

• Ri customer arrival (inflow) rate
• inter-arrival time 1/Ri
• Tp processing time
• Rp processing rate
• If we have one resource ? Rp 1/Tp
• In general when we have c c recourses, Rp c/Tp

5
A GAP Store

• Ri 6 customers per hour
• inter-arrival time 1/Ri 1/6 hour or 10
  minutes
• Tp processing time 5 minutes 5/60 1/12
  hour
• Rp processing rate
• If we have one resource ?

Rp 1/Tp 1/(1/12) 12 customers per hour
If we have c c recourses
Rp 2/Tp 12 customers per hour
6
Operational Performance Measures
Flow time T Ti Tp Inventory
I Ii Ip

• Ti waiting time in the inflow buffer
• Ii number of customers waiting in the inflow
  buffer

• Waiting time in the servers (processors)
• ?

7
Service Process Attributes

• ?? inflow rate / processing rate
• ?? throughout / process capacity
• ? R/ Rp lt 1
• Safety Capacity Rp R
• In the Gap example , R 6 per hour, processing
  time for a single server is 6 min ? Rp 12 per
  hour,
• ? R/ Rp 6/12 0.5
• Safety Capacity Rp R 12-6 6

8
Operational Performance Measures

• Given a single server. And a utilization of r
  0.4
• How many flow units are in the server ?

Given 2 servers. And a utilization of r
0.4 How many flow units are in the servers ?
9
Operational Performance Measures
Throughput R

• Flow time T Ti Tp
• Inventory I Ii Ip

I R? T Ii R? Ti
Ip R ? Tp R I/T Ii/Ti
Ip/Tp ? R/ Rp ? Ip / c
10
Operational Performance Measures

• I R? T Ii R? Ti
  Ip R ? Tp
• R I/T Ii/Ti
  Ip/Tp
• Tp ? if 1 server ? Rp 1/Tp
• In general, if c servers ? Rp c/Tp
• R Ip/Tp
• ? R/ Rp (Ip/Tp)/(c/Tp) Ip/c
• ? R/ Rp Ip/c

11
Financial Performance Measures

• Sales
• Throughput Rate
• Abandonment Rate
• Blocking Rate
• Cost
• Capacity utilization
• Number in queue / in system
• Customer service
• Waiting Time in queue /in system

12
Arrival Rate at an Airport Security Check Point
Customer Number Arrival Time Departure Time Time in Process
1 0 5 5
2 4 10 6
3 8 15 7
4 12 20 8
5 16 25 9
6 20 30 10
7 24 35 11
8 28 40 12
9 32 45 13
10 36 50 14
What is the queue size? What is the capacity
utilization?
13
Flow Times with Arrival Every 6 Secs
Customer Number Arrival Time Departure Time Time in Process
1 0 5 5
2 6 11 5
3 12 17 5
4 18 23 5
5 24 29 5
6 30 35 5
7 36 41 5
8 42 47 5
9 48 53 5
10 54 59 5











What is the queue size? What is the capacity
utilization?
14
Effect of Variability
Customer Number Arrival Time Processing Time Time in Process
1-A 0 7 7
2-B 10 1 1
3-C 20 7 7
4-D 22 2 7
5-E 32 8 8
6-F 33 7 14
7-G 36 4 15
8-H 43 8 16
9-I 52 5 12
10-J 54 1 11
What is the queue size? What is the capacity
utilization?
15
Effect of Synchronization
Customer Number Arrival Time Processing Time Time in Process
1-E 0 8 8
2-H 10 8 8
3-D 20 2 2
4-A 22 7 7
5-B 32 1 1
6-J 33 1 1
7-C 36 7 7
8-F 43 7 7
9-G 52 4 4
10-I 54 5 7
What is the queue size? What is the capacity
utilization?
16
Conclusion

• If inter-arrival and processing times are
  constant, queues will build up if and only if the
  arrival rate is greater than the processing rate
• If there is (unsynchronized) variability in
  inter-arrival and/or processing times, queues
  will build up even if the average arrival rate is
  less than the average processing rate
• If variability in interarrival and processing
  times can be synchronized (correlated), queues
  and waiting times will be reduced

17
Causes of Delays and Queues

• High, unsynchronized variability in
• - Interarrival times
• - Processing times
• High capacity utilization ? R/ Rp or low safety
  capacity
• Rs R - Rp due to
• - High inflow rate R
• - Low processing rate Rpc / Tp, which may be
  due to small-scale c and/or slow speed 1 / Tp

18
Drivers of Process Performance

• Two key drivers of process performance, (1)
  Interarrival time and processing time
  variability, and (2) Capacity utilization
• Variability in the interarrival and processing
  times can be measured using standard deviation.
• Higher standard deviation means greater
  variability.
• Coefficient of Variation the ratio of the
  standard deviation of interarrival time (or
  processing time) to the mean.
• Ci coefficient of variation for interarrival
  times
• Cp coefficient of variation for processing
  times

19
The Queue Length Formula
Utilization effect
Variability effect
x
???? Ri / Rp, where Rp c / Tp Ci and Cp are the
Coefficients of Variation (Standard
Deviation/Mean) of the inter-arrival and
processing times (assumed independent)
20
Factors affecting Queue Length

• This part factor captures the capacity
  utilization effect, which shows that queue length
  increases rapidly as the capacity utilization p
  increases to 1.

The second factor captures the variability
effect, which shows that the queue length
increases as the variability in interarrival and
processing times increases. Whenever there is
variability in arrival or in processing queues
will build up and customers will have to wait,
even if the processing capacity is not fully
utilized.
21
Throughput- Delay Curve
22
Example 8.4
A sample of 10 observations on Interarrival times
in seconds

• 10,10,2,10,1,3,7,9, 2, 6
• AVERAGE () ? Avg. interarrival time 6
• Ri 1/6 arrivals / sec.
• STDEV() ? Std. Deviation 3.94
• Ci 3.94/6 0.66
• C2i (0.66)2 0.4312

23
Example 8.4
A sample of 10 observations on processing times
in seconds

• 7,1,7 2,8,7,4,8,5, 1
• Tp 5 seconds
• Rp 1/5 processes/sec.
• Std. Deviation 2.83
• Cp 2.83/5 0.57
• C2p (0.57)2 0.3204

24
Example 8.4
Ri 1/6 lt RP 1/5 ? R Ri ? R/ RP
(1/6)/(1/5) 0.83 With c 1, the average number
of passengers in queue is as follows Ii
(0.832)/(1-0.83) (0.6620.572)/2 1.56 On
average 1.56 passengers waiting in line, even
though safety capacity is Rs RP - Ri 1/5 -
1/6 1/30 passenger per second, or 2 per minutes
25
Example 8.4

• Other performance measures
• TiIi/R (1.56)(6) 9.4 seconds
• Since TP 5 ? T Ti TP 14.4 seconds
• Total number of passengers in the process is
• I RT (1/6) (14.4) 2.4
• C2 ? Rp 2/5 ? ? (1/6)/(2/5) 0.42 ? Ii
  0.08

c ? Rs Ii Ti T I
1 0.83 0.03 1.56 9.38 14.38 2.4
2 0.42 0.23 0.08 0.45 5.45 0.91
26
Exponential Model

• In the exponential model, the interarrival and
  processing times are assumed to be independently
  and exponentially distributed with means 1/Ri and
  Tp.
• Independence of interarrival and processing times
  means that the two types of variability are
  completely unsynchronized.
• Complete randomness in interarrival and
  processing times.
• Exponentially distribution is Memoryless
  regardless of how long it takes for a person to
  be processed we would expect that person to spend
  the mean time in the process before being
  released.

27
The Exponential Model

• Poisson Arrivals
• Infinite pool of potential arrivals, who arrive
  completely randomly, and independently of one
  another, at an average rate Ri ? constant over
  time
• Exponential Processing Time
• Completely random, unpredictable, i.e., during
  processing, the time remaining does not depend on
  the time elapsed, and has mean Tp
• Computations
• Ci Cp 1
• K 8 , use Ii Formula
• K lt 8 , use Performance.xls

28
Example

• Interarrival time 6 secs ? Ri 10/min
• Tp 5 secs ? Rp 12/min for 1 server and 24
  /min for 2 servers
• Rs 12-10 2

c ? Rs Ii? Formula Ti Ri / Ii T Ti 5/60 I Ii c ?
1 0.83 2 4.16 0.42 0.5 5
2 0.42 14 0.18 0.02 0.1 1
29
t t in Exponential Distribution

• Mean inter-arrival time 1/Ri
• Probability that the time between two arrivals t
  is less than or equal to a specific vaule of t
• P(t t) 1 - e-Rit, where e 2.718282, the
  base of the natural logarithm
• Example 8.5
• If the processing time is exponentially
  distributed with a mean of 5 seconds, the
  probability that it will take no more than 3
  seconds is 1- e-3/5 0.451188
• If the time between consecutive passenger
  arrival is exponentially distributed with a mean
  of 6 seconds ( Ri 1/6 passenger per second)
• The probability that the time between two
  consecutive arrivals will exceed 10 seconds is
  e-10/6 0.1888

30
Performance Improvement Levers

• Decrease variability in customer inter-arrival
  and processing times.
• Decrease capacity utilization.
• Synchronize available capacity with demand.

31
Variability Reduction Levers

• Customers arrival are hard to control
• Scheduling, reservations, appointments, etc.
• Variability in processing time
• Increased training and standardization processes
• Lower employee turnover rate more experienced
  work force
• Limit product variety

32
Capacity Utilization Levers

• If the capacity utilization can be decreased,
  there will also be a decrease in delays and
  queues.
• Since ?Ri/RP, to decrease capacity utilization
  there are two options
• Manage Arrivals Decrease inflow rate Ri
• Manage Capacity Increase processing rate RP
• Managing Arrivals
• Better scheduling, price differentials,
  alternative services
• Managing Capacity
• Increase scale of the process (the number of
  servers)
• Increase speed of the process (lower processing
  time)

33
Synchronizing Capacity with Demand

• Capacity Adjustment Strategies
• Personnel shifts, cross training, flexible
  resources
• Workforce planning season variability
• Synchronizing of inputs and outputs

34
Effect of Pooling
Ri/2
Server 1
Queue 1
Ri
Ri/2
Server 2
Queue 2
Server 1
Ri
Queue
Server 2
35
Effect of Pooling

• Under Design A,
• We have Ri 10/2 5 per minute, and TP 5
  seconds, c 1 and K 50, we arrive at a total
  flow time of 8.58 seconds
• Under Design B,
• We have Ri 10 per minute, TP 5 seconds, c2 and
  K50, we arrive at a total flow time of 6.02
  seconds
• So why is Design B better than A?
• Design A the waiting time of customer is
  dependent on the processing time of those ahead
  in the queue
• Design B, the waiting time of customer is only
  partially dependent on each preceding customers
  processing time
• Combining queues reduces variability and leads to
  reduce waiting times

36
Effect of Buffer Capacity

• Process Data
• Ri 20/hour, Tp 2.5 mins, c 1, K Lines
  c
• Performance Measures

K 4 5 6
Ii 1.23 1.52 1.79
Ti 4.10 4.94 5.72
Pb 0.1004 0.0771 0.0603
R 17.99 18.46 18.79
r 0.749 0.768 0.782
37
Economics of Capacity Decisions

• Cost of Lost Business Cb
• / customer
• Increases with competition
• Cost of Buffer Capacity Ck
• /unit/unit time
• Cost of Waiting Cw
• /customer/unit time
• Increases with competition
• Cost of Processing Cs
• /server/unit time
• Increases with 1/ Tp
• Tradeoff Choose c, Tp, K
• Minimize Total Cost/unit time
• Cb Ri Pb Ck K Cw I (or Ii) c Cs

38
Optimal Buffer Capacity

• Cost Data
• Cost of telephone line 5/hour, Cost of server
  20/hour, Margin lost 100/call, Waiting cost
  2/customer/minute
• Effect of Buffer Capacity on Total Cost

K 5(K c) 20 c 100 Ri Pb 120 Ii TC (/hr)
4 25 20 200.8 147.6 393.4
5 30 20 154.2 182.6 386.4
6 35 20 120.6 214.8 390.4
39
Optimal Processing Capacity
c K 6 c Pb Ii TC (/hr) 20c 5(Kc) 100Ri Pb 120 Ii
1 5 0.0771 1.542 386.6
2 4 0.0043 0.158 97.8
3 3 0.0009 0.021 94.2
4 2 0.0004 0.003 110.8
40
Performance Variability

• Effect of Variability
• Average versus Actual Flow time
• Time Guarantee
• Promise
• Service Level
• P(Actual Time ? Time Guarantee)
• Safety Time
• Time Guarantee Average Time
• Probability Distribution of Actual Flow Time
• P(Actual Time ? t) 1 EXP(- t / T)

41
Effect of Blocking and Abandonment

• Blocking the buffer is full new arrivals are
  turned away
• Abandonment the customers may leave the process
  before being served
• Proportion blocked Pb
• Proportion abandoning Pa

42
Net Rate Ri(1- Pb)(1- Pa)Throughput
RateRminRi(1- Pb)(1- Pa),Rp
Effect of Blocking and Abandonment
43
Example 8.8 - DesiCom Call Center

• Arrival Rate Ri 20 per hour0.33 per min
• Processing time Tp 2.5 minutes (24/hr)
• Number of servers c1
• Buffer capacity K5
• Probability of blocking Pb0.0771
• Average number of calls on hold Ii1.52
• Average waiting time in queue Ti4.94 min
• Average total time in the system T7.44 min
• Average total number of customers in the system
  I2.29

44
Example 8.8 - DesiCom Call Center

• Throughput Rate
• RminRi(1- Pb),Rp min20(1-0.0771),24
• R18.46 calls/hour
• Server utilization
• R/ Rp18.46/240.769

45
Example 8.8 - DesiCom Call Center
Number of lines 5 6 7 8 9 10
Number of servers c 1 1 1 1 1 1
Buffer Capacity K 4 5 6 7 8 9
Average number of calls in queue 1.23 1.52 1.79 2.04 2.27 2.47
Average wait in queue Ti (min) 4.10 4.94 5.72 6.43 7.08 7.67
Blocking Probability Pb () 10.04 7.71 6.03 4.78 3.83 3.09
Throughput R (units/hour) 17.99 18.46 18.79 19.04 19.23 19.38
Resource utilization .749 .769 .782 .793 .801 .807
46
Capacity Investment Decisions

• The Economics of Buffer Capacity
• Cost of servers wages 20/hour
• Cost of leasing a telephone line5 per line per
  hour
• Cost of lost contribution margin 100 per
  blocked call
• Cost of waiting by callers on hold 2 per minute
  per customer
• Total Operating Cost is 386.6/hour

47
Example 8.9 - Effect of Buffer Capacity on Total
Cost
Number of lines n 5 6 7 8 9
Number of CSRs c 1 1 1 1 1
Buffer capacity Kn-c 4 5 6 7 8
Cost of servers (/hr)20c 20 20 20 20 20
Cost of tel.lines (/hr)5n 25 30 35 40 45
Blocking Probability Pb () 10.04 7.71 6.03 4.78 3.83
Lost margin 100RiPb 200.8 154.2 120.6 95.6 76.6
Average number of calls in queue Ii 1.23 1.52 1.79 2.04 2.27
Hourly cost of waiting120Ii 147.6 182.4 214.8 244.8 272.4
Total cost of service, blocking and waiting (/hr) 393.4 386.6 390.4 400.4 414
48
Example 8.10 - The Economics of Processing
Capacity

• The number of line is fixed n6
• The buffer capacity K6-c

c K Blocking Pb() Lost Calls RiPb (number/hr) Queue length Ii Total Cost (/hour)
1 5 7.71 1.542 1.52 3020(1.542x100)(1.52x120)386.6
2 4 0.43 0.086 0.16 3040(0.086x100)(0.16x120)97.8
3 3 0.09 0.018 0.02 3060(0.018x100)(0.02x120)94.2
4 2 0.04 0.008 0.00 3080(0.008x100)(0.00x120)110.8
49
Variability in Process Performance

• Why considering the average queue length and
  waiting time as performance measures may not be
  sufficient?
• Average waiting time includes both customers
  with very long wait and customers with short or
  no wait.
• We would like to look at the entire probability
  distribution of the waiting time across all
  customers.
• Thus we need to focus on the upper tail of the
  probability distribution of the waiting time, not
  just its average value.

50
Example 8.11 - WalCo Drugs

• One pharmacist, Dave
• Average of 20 customers per hour
• Dave takes Average of 2.5 min to fill
  prescription
• Process rate 24 per hour
• Assume exponentially distributed interarrival and
  processing time we have single phase, single
  server exponential model
• Average total process is
• T 1/(Rp Ri) 1/(24 -20) 0.25 or 15 min

51
Example 8.11 - Probability distribution of the
actual time customer spends in process (obtained
by simulation)
52
Example 8.11 - Probability Distribution Analysis

• 65 of customers will spend 15 min or less in
  process
• 95 of customers are served within 40 min
• 5 of customers are the ones who will bitterly
  complain. Imagine if they new that the average
  customer spends 15 min in the system.
• 35 may experience delays longer than Average
  T,15min

53
Service PromiseTduedate , Service Level
Safety Time

• SL The probability of fulfilling the stated
  promise. The Firm will set the SL and calculate
  the Tduedate from the probability distribution of
  the total time in process (T).
• Safety time is the time margin that we should
  allow over and above the expected time to deliver
  service in order to ensure that we will be able
  to meet the required date with high probability
• Tduedate T Tsafety
• Prob(Total time in process lt Tduedate) SL
• Larger SL results in grater probability of
  fulfilling the promise.

54
Due Date Quotation

• Due Date Quotation is the practice of promising a
  time frame within which the product will be
  delivered.
• We know that in single-phase single server
  service process the Actual total time a customer
  spends in the process is exponentially
  distributed with mean T.
• SL Prob(Total time in process lt Tduedate) 1
  EXP( - Tduedate /T)
• Which is the fraction of customers who will no
  longer be delayed more than promised.
• Tduedate -T ln(1 SL)

55
Example 8.12 - WalCo Drug

• WalCo has set SL 0.95
• Assuming total time for customers is exponential
• Tduedate -T ln(1 SL)
• Tduedate -T ln(0.05) 3T
• Flow time for 95 percentile of exponential
  distribution is three times the average T
• Tduedate 3 15 45
• 95 of customers will get served within 45 min
• Tduedate T Tsafety
• Tsafety 45 15 30 min
• 30 min is the extra margin that WalCo should
  allow as protection against variability

56
Relating Utilization and Safety Time Safety
Time Vs. Capacity Utilization

• Capacity utilization ? 60
  70 80 90
• Waiting time Ti 1.5Tp
  2.33Tp 4Tp
  9Tp
• Total flow time T Ti Tp
  2.5Tp 3.33Tp 5Tp 10Tp
• Promised time Tduedate 7.7Tp
  10Tp 15Tp 30Tp
• Safety time Tsafety Tduedate T 5Tp
  6.67Tp 10Tp 20Tp
• Higher the utilization Longer the promised time
  and Safety time
• Safety Capacity decreases when capacity
  utilization increases
• Larger safety capacity, the smaller safety time
  and therefore we can promise a shorter wait

57
Managing Customer Perceptions and Expectations

• Uncertainty about the length of wait (Blind
  waits) makes customers more impatient.
• Solution is Behavioral Strategies
• Making the waiting customers comfortable
• Creating distractions
• Offering entertainment

58
Thank you

• Questions?